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$L$-functions and linear periods for $\mathbf{GL}_4 \times \mathbf{GL}_2$ and $\mathbf{GU}_{2,2}\times \mathbf{GL}_2$

Antonio Cauchi, Armando Gutierrez Terradillos

TL;DR

The paper introduces a novel integral representation for the $L$-function $L(s, \wedge^2 \otimes \mathrm{std}_2)$ attached to generic cusp forms on ${\mathbf{GU}}_{2,2}\times{\mathbf{GL}}_2$ and ${\mathbf{GL}}_4\times{\mathbf{GL}}_2$, and uses it to relate central $L$-values to both non-spherical and spherical automorphic periods. It deploys a two-variable zeta integral built from a Klingen Eisenstein series and leverages theta correspondences, Garrett’s pullback, and Siegel–Weil theory for similitudes to connect $L(\tfrac{1}{2},\pi\otimes\sigma,\wedge^2\otimes\mathrm{std}_2)$ with periods on $\mathbf{GL}_2\times\mathbf{GL}_2$ and related groups. The archimedean analysis, local factor computations (in inert and split cases), and the seesaw identities culminate in two main results: a non-spherical degree-12 central value relation and a degree-20 spherical period relation, with the unramified GL$_4$×GL$_2$ case providing evidence toward Wan–Zhang. These constructions illuminate a cohesive framework linking Langlands $L$-values to periods via relative trace formula techniques and theta correspondences, with potential arithmetic applications in cohomology and automorphic $L$-value conjectures.

Abstract

We give a new integral representation of the $\wedge^2 \otimes \mathrm{std}_2$ $L$-function of generic cusp forms on $\mathbf{GU}_{2,2}\times \mathbf{GL}_2$ and $\mathbf{GL}_4 \times \mathbf{GL}_2$ and we use it to prove a relation between its central $L$-value and the non-spherical period over $\mathbf{GL}_2 \times_{\mathrm{det}} \mathbf{GL}_2$. Exploiting the theta correspondence for $(\mathbf{GL}_4,\mathbf{GL}_4)$, we obtain a relation between the central value of the $L$-function attached to the strongly tempered spherical pair $(\mathbf{GL}_4 \times \mathbf{GL}_2,\mathbf{GL}_2 \times \mathbf{GL}_2)$ and its associated period. In the case of cusp forms on $\mathbf{GL}_4 \times \mathbf{GL}_2$ that are unramified everywhere, our formula gives new evidence towards a conjecture of Wan-Zhang.

$L$-functions and linear periods for $\mathbf{GL}_4 \times \mathbf{GL}_2$ and $\mathbf{GU}_{2,2}\times \mathbf{GL}_2$

TL;DR

The paper introduces a novel integral representation for the -function attached to generic cusp forms on and , and uses it to relate central -values to both non-spherical and spherical automorphic periods. It deploys a two-variable zeta integral built from a Klingen Eisenstein series and leverages theta correspondences, Garrett’s pullback, and Siegel–Weil theory for similitudes to connect with periods on and related groups. The archimedean analysis, local factor computations (in inert and split cases), and the seesaw identities culminate in two main results: a non-spherical degree-12 central value relation and a degree-20 spherical period relation, with the unramified GL×GL case providing evidence toward Wan–Zhang. These constructions illuminate a cohesive framework linking Langlands -values to periods via relative trace formula techniques and theta correspondences, with potential arithmetic applications in cohomology and automorphic -value conjectures.

Abstract

We give a new integral representation of the -function of generic cusp forms on and and we use it to prove a relation between its central -value and the non-spherical period over . Exploiting the theta correspondence for , we obtain a relation between the central value of the -function attached to the strongly tempered spherical pair and its associated period. In the case of cusp forms on that are unramified everywhere, our formula gives new evidence towards a conjecture of Wan-Zhang.
Paper Structure (52 sections, 58 theorems, 381 equations)

This paper contains 52 sections, 58 theorems, 381 equations.

Table of Contents

  1. Introduction
  2. Main Theorems
  3. New integral representations
  4. Sketch of the proofs of Theorems \ref{['Thma']} and \ref{['Thmb']}
  5. Preliminaries
  6. Notation
  7. Symplectic and Unitary Groups
  8. Maximal parabolic subgroups
  9. Orthogonal groups
  10. Maximal parabolic subgroups
  11. Quaternionic quadratic spaces
  12. The Weil representation for similitude groups
  13. Dual groups and the $\wedge^2 \otimes {\rm std}_2$ $L$-function
  14. Exterior square representations
  15. $L$-functions
  16. ...and 37 more sections

Key Result

Theorem 1

Let $\pi$ and $\sigma$ be cuspidal automorphic representations of $\mathbf{GL}_4$ and $\mathbf{GL}_2$ respectively, which are unramified everywhere and such that the product of their central characters is trivial. Then, where $L(s, \pi \otimes \sigma, \wedge^2 \otimes \mathrm{std}_2)$ denotes the completed $L$-function.

Theorems & Definitions (120)

  • Theorem 1
  • Conjecture 1.1: Wan:Zhang:Spherical:Periods, SakellaridisVenkatesh
  • Theorem 2
  • Theorem 3: Theorem \ref{['zetaintegralfinalthm']}
  • Corollary 4
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • ...and 110 more